Bivariate Lagrange interpolation at the node points of non-degenerate Lissajous curves
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Motivated by an application in Magnetic Particle Imaging, we study bivariate Lagrange interpolation at the node points of Lissajous curves. The resulting theory is a generalization of the polynomial interpolation theory developed for a node set known as Padua points. With appropriately defined polynomial spaces, we will show that the node points of non-degenerate Lissajous curves allow unique interpolation and can be used for quadrature rules in the bivariate setting. An explicit formula for the Lagrange polynomials allows to compute the interpolating polynomial with a simple algorithmic scheme. Compared to the already established schemes of the Padua and Xu points, the numerical results for the proposed scheme show similar approximation errors and a similar growth of the Lebesgue constant.
Mathematics Subject Classification41A05 41A10 42B05 65T40
The authors gratefully acknowledge the financial support of the German Federal Ministry of Education and Research (BMBF, Grant number 13N11090), the German Research Foundation (DFG, Grant number BU 1436/9-1 and ER 777/1-1), the European Union and the State Schleswig-Holstein (EFRE, grant number 122-10-004).
- 17.Grüttner, M., Knopp, T., Franke, J., Heidenreich, M., Rahmer, J., Halkola, A., Kaethner, C., Borgert, J., Buzug, T.M.: On the formulation of the image reconstruction problem in magnetic particle imaging. Biomed. Tech. Biomed. Eng. 58(6), 583–591 (2013)Google Scholar
- 19.Kaethner, C., Ahlborg, M., Bringout, G., Weber, M., Buzug, T.M.: Axially elongated field-free point data acquisition in magnetic particle imaging. IEEE Trans. Med. Imag. 34(2), 381–387 (2015)Google Scholar